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Theorem cnvcnvss 5452
Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
cnvcnvss  |-  `' `' A  C_  A

Proof of Theorem cnvcnvss
StepHypRef Expression
1 cnvcnv 5450 . 2  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
2 inss1 3711 . 2  |-  ( A  i^i  ( _V  X.  _V ) )  C_  A
31, 2eqsstri 3527 1  |-  `' `' A  C_  A
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3106    i^i cin 3468    C_ wss 3469    X. cxp 4990   `'ccnv 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000
This theorem is referenced by:  funcnvcnv  5637  foimacnv  5824  cnvfi  7793  structcnvcnv  14490  strlemor1  14571  mvdco  16259  fcoinver  27119  fcnvgreu  27172  cnvct  27196  cnvtrrel  36667
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